Multidimensional Detonation Solutions from Reactive Navier-Stokes Equations (original) (raw)

Multidimensional Detonation Solutions from Reactive Navier-Stokes Equations 1

2008

Detonation solutions from reactive Navier-Stokes equations

37th Aerospace Sciences Meeting and Exhibit, 1999

This study will describe multi-dimensional detonation wave solutions of the compressible reactive Navier-Stokes equations. As discussed in detail by Fickett and Davis [1], a steady onedimensional detonation with a spatially resolved reaction zone structure is known as ZND wave, named after Zeldovich, von Neumann, and Döring. In experiments [2] and calculations with simplified models [3], [4], [5], [6], it has been observed and predicted that these ZND waves are unstable. In the experiments, detonation in a tube with walls coated with a thin layer of soot etches detailed regular patterns on the tube walls, indicating the existence of cellular detonation wave structure. Linear analysis [3] demonstrates the fundamental instability of the one-dimensional ZND structure. This is extended by analysis of the full one-dimensional unsteady Euler equations to describe galloping detonations [4]. In two-dimensional calculations [5] it is found that complex cellular structures and transverse waves evolve from the original one-dimensional, steady detonation, for cases in which the steady one-dimensional structure is unstable. Grismer and Powers [6] have shown numerically that detonations which are guaranteed stable in one dimension can be unstable when the geometry is relaxed to include two-dimensional effects. Most calculations are done with compressible reactive Euler equations, and two-dimensional cell size is often predicted to be dependent on grid resolution, which indicates numerical viscosity is playing a determining role in predicting the physics. To remedy this, we reintroduce in this study the usually-neglected physical mechanisms of mass, momentum, and energy diffusion to the conservation equations. In this abstract, we give results of our initial calculations which are very similar to those of Lindström [7]. The full paper will extend these results to consider the effects of diffusion on one-dimensional structure, wall boundary layer effects, and the corrections for diffusion coefficients with dependency on thermodynamic properties.

A computational study of the interaction of gaseous detonations with a compressible layer

Physics of Fluids, 2017

The propagation of two-dimensional cellular gaseous detonation bounded by an inert layer is examined via computational simulations. The analysis is based on the high-order integration of the reactive Euler equations with a one-step irreversible reaction. To assess whether the cellular instabilities have a significant influence on a detonation yielding confinement, we achieved numerical simulations for several mixtures from very stable to mildly unstable. The cell regularity was controlled through the value of the activation energy, while keeping constant the ideal Zel’dovich - von Neumann - Döring (ZND) half-reaction length. For stable detonations, the detonation velocity deficit and structure are in accordance with the generalized ZND model, which incorporates the losses due to the front curvature. The deviation with this laminar solution is clear as the activation energy is more significant, increasing the flow field complexity, the variations of the detonation velocity, and the t...

Hydrodynamic instabilities and transverse waves in propagation mechanism of gaseous detonations

Acta Astronautica, 2013

The present study examines the role of transverse waves and hydrodynamic instabilities mainly, Richtmyer-Meshkov instability (RMI) and Kelvin-Helmholtz instability (KHI) in detonation structure using two-dimensional high-resolution numerical simulations of Euler equations. To compare the numerical results with those of experiments, Navier-Stokes simulations are also performed by utilizing the effect of diffusion in highly irregular detonations. Results for both moderate and low activation energy mixtures reveal that upon collision of two triple points a pair of forward and backward facing jets is formed. As the jets spread, they undergo Richtmyer-Meshkov instability. The drastic growth of the forward jet found to have profound role in re-acceleration of the detonation wave at the end of a detonation cell cycle. For irregular detonations, the transverse waves found to have substantial role in propagation mechanism of such detonations. In regular detonations, the lead shock ignites all the gases passing through it, hence, the transverse waves and hydrodynamic instabilities do not play crucial role in propagation mechanism of such regular detonations. In comparison with previous numerical simulations present simulation using single-step kinetics shows a distinct keystone-shaped region at the end of the detonation cell.

Numerical study of 3D gaseous detonations in a square channel

Aerotecnica Missili & Spazio, 2018

The multidimensional structure of mildly unstable detonations are examined by numerical computations. These phenomenon have grown in interest since the development of propulsion devices such as pulsed and rotating detonation engines. Rectangular, diagonal and spinning modes are observed in a near-limit propagation detonation. High-order numerical integration of the reactive Euler equations have been performed to analyze the averaged structure, the shock dynamics of a single-cell detonation propagating in a square channel. Computations show a good agreement with the experimental cellular structure, showing the relevance of the slapping waves in the rectangular modes. The hydrodynamic thickness as well as the pdf shock dynamics are similar in the 2D and 3D cases, but the mean quantities vary on a quantitative basis. Moreover, the presence of strong forward jets is attested, which comes from simultaneous triple point line collisions with the walls.

The hydrodynamic structure of detonations

20th ICDERS meeting, …, 2005

Most detonations are unstable, their reaction zones are turbulent and their structure departs significantly from the idealized one-dimensional Zeldovich-Von Neumann-Doering model (ZND). Recent numerical studies further demonstrated that detonation waves are chaotic, following the Feigenbaum bifurcation route. Parameters corresponding to typical reacting systems fall in the chaotic regime. Since a deterministic theory for such detonations is not possible, the present study considers a stochastic one-dimensional treatment and model for such detonation waves. Real and numerical experiments are used to verify whether the space and time-averaged structure of detonations can be described by a generalized probabilistic onedimensional ZND theory, with a statistically determined sonic surface, dictated by the competition between the various global chemical, mechanical and thermal relaxation processes.

Cell-like structure of unstable oblique detonation wave from high-resolution numerical simulation

Proceedings of the Combustion Institute, 2007

A comprehensive numerical study was carried out to investigate the unsteady cell-like structures of oblique detonation waves (ODWs) for a fixed Mach 7 inlet flow over a wedge of 30°turning angle. The effects of grid resolution and activation energy were examined systematically at a dimensionless heat addition of 10. The ODW front remains stable for a low activation energy regardless of grid resolution, but becomes unstable for a high activation energy featuring a cell-like wave front structure. Similar to the situation with an ordinary normal detonation wave (NDW), a continuous increase in the activation energy eventually causes the wave-front oscillation to transit from a regular to an irregular pattern. The wave structure of an unstable ODW, however, differs considerably from that of a NDW. Under the present flow condition, triple points and transverse waves propagate downstream, and the numerical smoke-foil record exhibits traces of triple points that rarely intersect with each other. Several instability-driving mechanisms were conjectured from the highly refined results. Since the reaction front behind a shock wave can be easily destabilized by disturbance inherent in the flowfield, the ODW front becomes unstable and displays cell-like structures due to the local pressure oscillations and/or the reflected shock waves originating from the triple points. The combined effects of various instability sources give rise to a highly unstable and complex flow structure behind an unstable ODW front.

Diffusion and hydrodynamic instabilities in gaseous detonations

Combustion and Flame, 2012

To clarify the role played by diffusion in detonation structure, two-dimensional numerical simulations are performed by solving the Navier-Stokes equations and considering the single step Arrhenius kinetic as reaction model. The effect of diffusion on the generation of vortices produced by hydrodynamic instabilities (Richtmyer-Meshkov (RM) and Kelvin Helmholtz (KH) instabilities) is investigated. Mixtures with both low and high activation energies, characterized by their regular and irregular detonation structures, are considered. The computations are performed with resolutions ranging from 25 to 10 3 cells per half reaction length of the ZND structure. Resolution studies of the Navier-Stokes solution for irregular detonations in moderate activation energy mixtures shows that to capture a proper structure, to be at least in qualitative agreement with experimental observations, resolution more that 300 cells per half reaction length is required. However, in mixtures with low activation energy a resolution of 25 cells per half reaction length gives a reasonable physical structure of the detonation. Results provided by very high resolution for irregular structure detonations reveal that the major effect of diffusion occurs at shear layers and unburned pockets boundaries. Diffusion suppresses the small-scale vortices produced by KH instabilities and decreases the turbulent mixing rate of burned and partly burned gases at shear layers. However, behind the shock front, where less concentration of small-scale vortices exist, the diffusion of heat and mass from neighboring hot regions of burned material to the unreacted gases increases the burning rate of the un-reacted pockets. Comparison of the structure obtained by solving the Euler equations with the solution of the Navier-Stokes equations shows that, the strength of the shock front in Navier-Stokes solution is higher than that in Euler solution. Due to the absence of hydrodynamic instabilities behind the main front of regular structure detonations, the results obtained by solving the Euler equations and Navier-Stokes equations are similar for detonations with regular structure even in high resolution simulations.

On the dynamics of multi-dimensional detonation

Journal of Fluid Mechanics, 1996

We present an asymptotic theory for the dynamics of detonation when the radius of curvature of the detonation shock is large compared to the one-dimensional, steady, Chapman-Jouguet (CJ) detonation reaction-zone thickness. The analysis considers additional time-dependence in the slowly varying reaction zone to that considered in previous works. The detonation is assumed to have a sonic point in the reactionzone structure behind the shock, and is referred to as an eigenvalue detonation. A new, iterative method is used to calculate the eigenvalue relation, which ultimately is expressed as an intrinsic, partial differential equation (PDE) for the motion of the shock surface. Two cases are considered for an ideal equation of state. The first corresponds to a model of a condensed-phase explosive, with modest reaction rate sensitivity, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D,, the first normal time derivative of the normal shock velocity, D,,, and the shock curvature, IC. The second case corresponds to a gaseous explosive mixture, with the large reaction rate sensitivity of Arrhenius kinetics, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D,, its first and second normal time derivatives of the normal shock velocity, b,, B,, and the shock curvature, IC, and its first normal time derivative of the curvature, k. For the second case, one obtains a one-dimensional theory of pulsations of plane CJ detonation and a theory that predicts the evolution of self-sustained cellular detonation. Versions of the theory include the limits of near-CJ detonation, and when the normal detonation velocity is significantly below its CJ value. The curvature of the detonation can also be of either sign, corresponding to both diverging and converging geometries.

On the dynamics and linear stability of one-dimensional steady detonation waves

Journal of Physics A: Mathematical and Theoretical, 2012

A detailed analysis of the dynamics and linear stability of a steady one-dimensional detonation wave propagating in a binary reactive system with an Arrhenius chemical kinetics of type A + A ⇋ B + B is carried out. Starting from the frame of the kinetic theory, the binary reactive mixture is modelled at the mesoscopic scale by the reactive Boltzmann equation (BE), assuming hard sphere cross sections for elastic collisions and step cross sections with activation energy for reactive interactions. The corresponding hydrodynamic limit is based on a second-order non-equilibrium solution of the BE obtained in a previous paper, using the Chapman-Enskog method in a chemical regime for which the reactive interactions are less frequent than the elastic collisions. The resulting hydrodynamic governing equations are the reactive Euler equations, including a rate law which exhibits an explicit dependence on the reaction heat and forward activation energy of the chemical reaction. These equations are used to describe the spatial structure of the steady detonation wave solution and investigate how this structure varies with the reaction heat. The response of the steady solution to one-dimensional disturbances is studied using a normal mode linear approach which leads to an initial value problem for the state variable disturbances in the reaction zone. The stability problem is treated numerically, using an iterative shooting technique to determine the unstable modes. The analysis here developed emphasizes the influence of the chemical reaction heat and activation energy on the linear stability spectra.

Multidimensional Detonation Solutions from Reactive Navier-Stokes Equations (original) (raw)

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